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Properties

Label 2-1110-111.68-c1-0-9
Degree $2$
Conductor $1110$
Sign $-0.594 - 0.804i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.72 + 0.161i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (1.10 − 1.33i)6-s + 0.212·7-s + (0.707 + 0.707i)8-s + (2.94 − 0.556i)9-s − 1.00·10-s − 4.71·11-s + (0.161 + 1.72i)12-s + (1.41 − 1.41i)13-s + (−0.150 + 0.150i)14-s + (−1.33 − 1.10i)15-s − 1.00·16-s + (2.20 + 2.20i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.995 + 0.0931i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.451 − 0.544i)6-s + 0.0804·7-s + (0.250 + 0.250i)8-s + (0.982 − 0.185i)9-s − 0.316·10-s − 1.42·11-s + (0.0465 + 0.497i)12-s + (0.391 − 0.391i)13-s + (−0.0402 + 0.0402i)14-s + (−0.344 − 0.285i)15-s − 0.250·16-s + (0.534 + 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.594 - 0.804i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6413381267\)
\(L(\frac12)\) \(\approx\) \(0.6413381267\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.72 - 0.161i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-5.76 - 1.94i)T \)
good7 \( 1 - 0.212T + 7T^{2} \)
11 \( 1 + 4.71T + 11T^{2} \)
13 \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \)
17 \( 1 + (-2.20 - 2.20i)T + 17iT^{2} \)
19 \( 1 + (1.02 - 1.02i)T - 19iT^{2} \)
23 \( 1 + (-1.60 - 1.60i)T + 23iT^{2} \)
29 \( 1 + (-3.47 + 3.47i)T - 29iT^{2} \)
31 \( 1 + (-0.648 - 0.648i)T + 31iT^{2} \)
41 \( 1 + 1.68T + 41T^{2} \)
43 \( 1 + (-2.65 + 2.65i)T - 43iT^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 - 6.21iT - 53T^{2} \)
59 \( 1 + (5.08 + 5.08i)T + 59iT^{2} \)
61 \( 1 + (0.209 + 0.209i)T + 61iT^{2} \)
67 \( 1 - 6.62iT - 67T^{2} \)
71 \( 1 - 9.09iT - 71T^{2} \)
73 \( 1 - 7.72iT - 73T^{2} \)
79 \( 1 + (12.0 - 12.0i)T - 79iT^{2} \)
83 \( 1 - 1.76iT - 83T^{2} \)
89 \( 1 + (6.73 - 6.73i)T - 89iT^{2} \)
97 \( 1 + (10.7 - 10.7i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.15314196099594211013086363753, −9.559405700258365111476526035031, −8.241733670460889260348918980068, −7.70053305343014431789811359736, −6.70168474290232209006766427072, −5.88288442991199754191648337084, −5.35295200082317987731960175351, −4.29347659449623476735546793310, −2.76278939286208751708780856466, −1.20688167681815057586589123173, 0.43000240223141222419682140899, 1.77493718279812762314579188971, 3.02552462140693341205249107961, 4.50407102352632622694003510078, 5.18603353330290339515664788997, 6.14160792864693225007232744650, 7.12423358292213257609127745022, 7.929422920170050092839946321173, 8.833819875393441511350764479982, 9.806632457909241460833285163679

Graph of the $Z$-function along the critical line